This question comes up quite regularly. While in general, the elliptic integral for arc length of a parametric curve has no closed-form solution, the problem is tractable for a quadratic Bezier. The integral is quite involved. It is discussed in this post, including a reference to the solution.

Now, just because you have a formula for something does not necessarily mean you should always use it. There are several divisions in the equation and some quads. can result in near-zero divisors. There are other numerical issues, some of which can be exposed by exploring the Degrafa demo provided in the above post.

Computationally, the closed-form solution is close to a wash with numerical integration. While the latter does have some subtle issues, I tend to use the numerical approach as it works for all parameteric curves and all values of the natural parameter. The closed-form solution for the quad. only works for quadratic Beziers at t=1. The numerical method can be used for arc-length parameterization as well as the segment problem; that is, computing the arc length of a segment of a curve from t=t1 to t=t2.

If you are a calculus student, you should study the derivation of the closed-form formula as it’s a great example of integration 🙂